State variable (system theory)

The term state variable or state variable is used in systems theory to describe the system in the state space representation .

The state variables of the state space representation physically describe the energy content of the storage elements contained in a technical dynamic system . They mean e.g. B. voltage on a capacitor, current in an inductance, potential and kinetic energy shares in a spring-mass damping system . The number of state variables of the state vector is the dimension of the state space. Technical application: control engineering . ${\ displaystyle n}$ ${\ displaystyle x_ {1} (t) \ cdots x_ {n} (t)}$ ${\ displaystyle {\ underline {x}} (t)}$

State variables of the state space representation

The state space theory comes from the USA in the 1960s by the mathematician and Stanford University professor Rudolf E. Kalman .

A dynamic system is a delimited, time-dependent functional unit that interacts with the environment through its signal inputs and signal outputs. The system can be a mechanical body, an electrical network { network (electrical engineering) }, but also a biological process or a component of the economy .

The state space representation is a system description of a mostly technical system with several energy stores and at least one input variable and one output variable. The input variable can have the value zero, in this case the system with its initial values of the energy storage at time t = 0 for t> 0 is left to its own devices and, after a sufficiently long time, strives for the value zero with its output variable (in the case of a stable system , Example of damped spring pendulum ). From a physical point of view, the state of a dynamic system is determined by the energy content of the energy storage in the system. The state variables describe the energy content of the storage elements contained in the system.

Symbolic block diagram of a model of a transmission system with state variables in the state space representation for a single variable system.

{\ displaystyle x_ {i} (t)}

Two ways lead to a state space model of a dynamic system:

by establishing the physical interrelationship using differential equations of the first order,
by converting a differential equation of higher order into a system of coupled differential equations of the first order.

All relationships between the state variables, the input variables and the output variables of a state space model are represented in the form of matrices and vectors . The state space model is described by two equations - the state differential equation and the output equation.

The state variables can be read directly from the state space representation given the output equation (if n> m, the passage factor d = 0). ${\ displaystyle x_ {i} (t)}$

{\ displaystyle {\ begin {bmatrix} y (t) \\\ end {bmatrix}} = \ underbrace {\ begin {bmatrix} c_ {1} & ... & c_ {n} \\\ end {bmatrix}} _ {Output vector} \ \ cdot \ underbrace {\ begin {bmatrix} x_ {1} (t) \\ x_ {2} (t) \\\ vdots \\ x_ {n} (t) \\\ end {bmatrix}} _ {state vector}}

In the following, the creation, definition and use of the state variables that are combined in the state vector are dealt with. ${\ displaystyle x_ {i} (t)}$ ${\ displaystyle {\ underline {x}} (t)}$

Origin of the state variables

Typical physical quantities that as state variables such as currents , voltages , angle , paths , speeds , accelerations , forces , temperature to energy storage such as capacitances , inductances , masses and springs occur, giving the temporal behavior of the memory elements in response to the input signals or initial values again .

A linear dynamic system is described by an ordinary differential equation with constant coefficients.

{\ displaystyle a_ {n} \ cdot y ^ {(n)} (t) + \ \ cdots \ + a_ {2} \ cdot {\ ddot {y}} (t) + a_ {1} \ cdot {\ dot {y}} (t) + a_ {0} \ cdot y (t) = b_ {0} \ cdot u (t) + b_ {1} \ cdot {\ dot {u}} (t) + b_ { 2} \ cdot {\ ddot {u}} (t) + \ \ cdots \ + b_ {m} \ cdot u ^ {(m)} (t)}

The highest degree of derivation of reflects the number of storage elements in the system. This differential equation can be defined as a transfer function with the help of the Laplace transform : ${\ displaystyle y (t) \;}$

Transfer function of the polynomial representation and the decomposition into the pole-zero representation with real linear factors:

{\ displaystyle G (s) = {\ frac {Y (s)} {U (s)}} = {\ frac {b_ {m} s ^ {m} + \ ldots + b_ {2} s ^ {2 } + b_ {1} s + b_ {0}} {a_ {n} s ^ {n} + \ ldots + a_ {2} s ^ {2} + a_ {1} s + a_ {0}}}: = k \ cdot {\ frac {(s-s_ {n1}) (s-s_ {n2}) \ dotsm (s-s_ {nm})} {(s-s_ {p1}) (s-s_ {p2 }) \ dotsm (s-s_ {pn})}}}

Here m = number of zeros , n = number of poles , s = Laplace variable. ${\ displaystyle s_ {n}}$ ${\ displaystyle s_ {p}}$

State variables arise from the poles of the transmission system :

The state variables of a linear system of the nth order with n energy stores always arise from the poles of the transfer function . If the transmission system also has zeros - i.e. differentiating components - the state variables with the coefficients of the derivatives of the input variable u (t) are added to the output variable y (t). The poles of a transfer function determine the speed of the system movement and the stability. The zeros of a transfer function only affect the amplitudes of the system.

The state variables of a mathematical model of a dynamical system, a control system can be determined from the ordinary system of differential equations describing. The basis for solving the differential equation is the signal flow diagram with the graphic representation of the normal control form . The terms of the derivatives of the output variable y (t) are integrated and returned to the system input with the associated coefficients.

For each derivative of y (t) of the differential equation, the designation of the state variables x (t) is introduced as follows:

{\ displaystyle x_ {1} (t) = y (t)}

{\ displaystyle x_ {2} (t) = {\ frac {dy} {dt}} = {\ dot {y}}}

{\ displaystyle x_ {3} (t) = {\ ddot {y}}}

{\ displaystyle x_ {n} (t) = {\ frac {dy ^ {(n-1)}} {d ^ {(n-1)} t}} = y ^ {(n-1)} (t )}

All status variables are summarized in the status vector . At any point in time t ₍₀₎ , all information of the dynamic transmission system is contained in the state vector . ${\ displaystyle {\ underline {x}} (t)}$ ${\ displaystyle x_ {i} (t)}$ ${\ displaystyle {\ underline {x}} (t)}$

Block diagram of the signal flow diagram of a 3rd order transmission system in the normal control form.

Normal control form

In the description of states with normal forms , the equations of state take on particularly simple and useful forms for certain calculations. For the normal forms, the system description of the linear transfer system is based on the differential equation or the associated transfer function.

The signal structure of the normal control form presents itself as an analog time-continuous system which reproduces the solution of the differential equation with the input variable and at the same time shows the state variables . ${\ displaystyle u (t)}$ ${\ displaystyle y (t)}$ ${\ displaystyle x_ {1} (t), x_ {2} (t) \ cdots x_ {n} (t)}$

The block diagram of the normal control form shows the implementation and solution of the differential equation in the physical analog signal flows of the state variables including the output variable for a given input variable. The solution structure results from the transformation of the differential equation with the exemption of the highest derivative of y (t). All derivatives are in the order of the order by integrators integrated and fed back to the highest derivative with the corresponding coefficient and subtracted. The outputs of the integrators form the status variables. In addition to the status variables, y (t) is the system output variable. ${\ displaystyle a_ {n} \ cdot y ^ {(n)} (t)}$

The normal control form can be viewed as a further development of the method known in analog computing technology for solving an nth order differential equation with n integrators. If the coefficients of the state variables are known, the signal flows can be determined directly using numerical calculations for any input signals and displayed graphically.

If initial values of the system memory are available, these values can be set to the integrators of the signal flow diagram of the normal control form.

In the block diagram of the normal control form, the derivatives of are replaced by the state variables so that they no longer appear. ${\ displaystyle {\ tilde {y}} (t)}$ ${\ displaystyle x_ {1} \ cdots x_ {n}}$ ${\ displaystyle {\ tilde {y}} (t)}$

Interface of a controlled system in the state space.

Temporal behavior of the status variables in a controlled system

The time course of the state variables as a result of an input jump u (t) = 1 in the model shows the advantage of treating the system in the state space compared to a classic "output feedback" of the system. The state variables x (t) appear earlier than the output variable y (t). This behavior is used in the state control loop in that the state variables are traced back to a target / actual difference with the reference variable w (t).

Control systems generally consist of existing hardware and have at least one system input and one system output . If necessary, disturbance variables in various signal forms can attack at several intervention points . The system description is often described in the Laplace transformed s-domain as a transfer function. ${\ displaystyle u (t)}$ ${\ displaystyle y (t)}$ ${\ displaystyle d (t)}$ ${\ displaystyle G (s)}$

Step response of the status variables of a PT3 controlled system.

The specified hardware of the controlled system can also be described as a single-variable system in the state space as a function block. In this case the block has at least one system input , one system output and several outputs of the status variables and, if necessary, disturbance variable inputs . ${\ displaystyle u (t)}$ ${\ displaystyle y (t)}$ ${\ displaystyle x_ {i} (t)}$ ${\ displaystyle d (t)}$

The system behavior of the function block can be described by ordinary differential equations or transfer functions and converted into the normal control form of the state space representation. The integrators of the normal control form can assume the value zero for an observed point in time in the rest position, but they can also contain initial values for several derivatives. ${\ displaystyle t_ {0}}$ ${\ displaystyle y_ {0} ^ {(n)} (t)> 0}$

The differential equation or the transfer function of the controlled system is seldom known; the transfer behavior has to be identified . The status variables required for the controller must be determined by: ${\ displaystyle x_ {i} (t)}$

In practice, the status variables can be measured on the hardware of a controlled system, which is not always possible.

Observer by reconstructing the state variables if the route can be observed. The controlled system must be controllable .

All status variables must be available.

Pole-zero compensation in the state space is not allowed because information is lost.

State controller

Block diagram of a state controller for a 3rd order controlled system of a single variable system.

The pole assignment (pole specification) of the closed control loop is used as the design strategy for determining the evaluation factors of the state controller.

Simulations of a state control loop can easily be carried out with a good model of the controlled system on a programmable computer. The description of the signal flow diagram of the controlled system and the controller in the state space can be done in the form of matrices as well as with difference equations . Depending on the order of the differential equation, all state variables are fed to a state controller which acts on the input of the state space model of the controlled system. The feedback of all status variables creates a multi-loop control loop.

It is also possible to determine the gain factors of the state controller empirically. The controlled system in the normal control form and the state controller consisting of a multiple subtraction point with the prefilter V can be simulated on a personal computer by means of simulation using difference equations.

Because the integrators are connected in series as shown in the block diagram, only the state variable is a stationary variable> 0 if the input variable > 0 is constant. All other state variables - assuming a stable controlled system - tend towards the value zero. With the graphical representation of the system output variable of the simulation, the steady state for the transient response is given for a factor that has yet to be specified . The gain factors can be chosen as high as possible, e.g. B. 20-fold to 100-fold, however, it must be taken into account that the controlled system is in reality a hardware system that cannot accept arbitrarily high control values from the controller. ${\ displaystyle x1 (t)}$ ${\ displaystyle u (t)}$ ${\ displaystyle y (t)}$ ${\ displaystyle K_ {1} (x1)}$

Strategy for determining the gain factors in a controlled system with three state variables:

The reference variable is set to a standardized jump . ${\ displaystyle w (t)}$ ${\ displaystyle 1 (t)}$
The gain factors ; , and the pre-filter V are set to z. B. set to 20. ${\ displaystyle K_ {1} (x1)}$ ${\ displaystyle K_ {2} (x2)}$ ${\ displaystyle K_ {3} (x3)}$ ${\ displaystyle K_ {V}}$
The gain factors and are changed until the desired transient process z. B. takes place without overshoots. ${\ displaystyle K_ {2} (x2)}$ ${\ displaystyle K_ {3} (x3)}$
The steady state of is set to level 1 with the factor of the prefilter . ${\ displaystyle K_ {V}}$ ${\ displaystyle y (t)}$

The linear state controller evaluates the individual state variables of the controlled system with factors and adds the resulting state products to a target / actual value comparison.

{\ displaystyle u (t) = w (t) -x_ {1} \ cdot k_ {1} -x_ {2} \ cdot k_ {2} - \ cdots -x_ {n} \ cdot k_ {n} = w (t) - {\ underline {x}} (t) \ cdot {\ underline {k}} (t)}

This state controller is not a P controller, although such an impression could arise according to the signal flow diagram. The state variables with evaluation factors fed back with the controller run through the arithmetic circuit once again to solve the differential equation with n integrators and form new circular variables, which results in differentiating behavior. Therefore, depending on the magnitude of order n of the differential equation of the system, the effect of the fed back state variables corresponds to that of a controller. ${\ displaystyle PD _ {(n-1)}}$

The pole assignment ( pole specification ) of the closed control loop or empirically applies as a design strategy for determining the evaluation factors of the state controller . Because the integrators are connected in series, only the state variable x1 (t) = y (t) is a stationary variable> 0 if the input variable u (t) is constant. All other state variables - assuming a stable controlled system - tend towards the value zero.

A pre-filter before the target / actual comparison corrects the static error between w (t) and y (t), because this is a state feedback and not an output feedback . The pre-filter can be omitted if an additional PI controller is used instead of the pre-filter, which minimizes the static control deviation.

The control quality of a control with state variables cannot be achieved by any other control method.

Individual evidence

↑ State controller through pole specification. In: Oliver Nelles: Lecture manuscript for measurement and control technology II. University of Siegen, May 4, 2010.

literature

Jan Lunze: Control engineering 1: System theory basics, analysis and design of single-loop controls . 7th edition. Springer, 2008, ISBN 3-540-68907-9 .

H. Lutz, W. Wendt: Pocket book of control engineering with MATLAB and Simulink . Europa-Verlag, 2019, ISBN 978-3-8085-5869-0 .

[1] State controller through pole specification. In: Oliver Nelles: Lecture manuscript for measurement and control technology II. University of Siegen, May 4, 2010.